May I pose an alternative view to you.
From Day 1, Rugby League was linked with Unionism and the Labor Party. They had Labor politicians like Henry Hoyle, Edward O'Sullivan, Billy Hughes (future PM), Edward Larkin and James Joynton-Smith were all Labor men in prominent roles in the first 2 seasons of Rugby League in Australia. Ernest Broughton, the short-lived second President of the NSWRL, was the only Liberal in a prominent role.
The game for a long time has been proudly linked with the working class. It was built on the back of supporting the working class players who couldn't afford time off work due to injury.
Perhaps RL has never been that interested in trying to be in private schools, instead opting to stick with it's strong working class roots.
You've only got to look at recent news articles to see how passionate Labor is at (appearing to be) supporting RL
https://www.sbs.com.au/news/labor-will-back-png-team-in-nrl
https://www.melbournestorm.com.au/news/2018/09/17/labor-commits-to-future-of-victorian-rugby-league/
https://www.theroar.com.au/2009/10/24/john-howard-is-out-of-his-rugby-league/